Abstract
The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass of p-valent analytic functions. Many known results appear as special consequences of our work.
1. Introduction
Let denote the class of functions of the form which are analytic and multivalent in the open unit disk . Also let and denote the well-known classes of -valent starlike functions and -valent convex functions, respectively.
For given by (1) and given by the Hadamard product (or convolution) of and is given by Motivated by Ruscheweyh operator [1], Goel and Sohi [2] introduced a differential operator for -valent analytic functions given by with , and is a Pochhammer symbol given by It is obvious that when is any integer greater than , The following identity can be easily established: Using the generalized Ruscheweyh operator, we define a subclass of of -valent analytic functions as follows.
Definition 1. An analytic -valent function of the form (1) belongs to the class , if and only if
where , , , is real with , and .
By giving specific values to , , , , , and in , we obtain many important subclasses studied by various authors in earlier papers; see for details [3–6]; we list some of them as follows:(i) and , studied by Spacek [7] and Robertson [8], respectively; for the advancement work see [9–11];(ii) and , studied by both Owa et al. and Shams et al. [12, 13];(iii) and , introduced by Ravichandran et al. [14];(iv), considered by Latha [15];(v) and , the well-known classes of starlike and convex functions of order .From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.
We will assume throughout our discussion, unless otherwise stated, that , , , is real with , and .
2. Main Results
Theorem 2. Let with . Then , where .
Proof. Let . Then we obtain and this implies Also if , then we can easily obtain and this completes the proof.
Theorem 3. If , then and where is given by (5) and
Proof. Let . Then by Theorem 2, we have
Let us define by
Then is analytic in with and , . Let
Then (17) becomes
That is,
where is given by (15). Using (8) in (20), we obtain
or, equivalently,
Comparing the coefficients of on both sides,
Taking absolute on both sides and then applying the coefficient estimates for Caratheodory functions [3], we have
We apply mathematical induction on (24). So for ,
which shows that (13) is true. For ,
and using (13), we have
Therefore, (14) holds for .
Assume that (14) is true for ; that is,
Consider
Therefore, the result is true for , and hence by using mathematical induction, (14) holds true for all .
If we put , , , and in Theorem 3, we obtain the result proved in [12].
Corollary 4. If , then
If one takes in Corollary 4, one obtains the following inequality:
which was proved by Robertson [16].
By setting , , , and in Theorem 3, one obtains the result proved in [12].
Corollary 5. If , then Letting in Corollary 5, one gets the following inequality proved by Robertson [16]:
Theorem 6. If and satisfies where is given by (5), then .
Proof. Suppose (34) holds. Also let us suppose Then Using (8) and then simplifications gives Now consider The last expression is bounded by 1 if and this completes the proof.
For , , , and in Theorem 6, we obtain the following.
Corollary 7. If and satisfies
then , the class of -valent starlike functions of order .
For , , , and in Theorem 6, one has the following.
Corollary 8. If and satisfies
then , the class of -valent convex functions of order .
Further for in both the last two corollaries, one obtains the results for the classes and which was proved by Merkes et al. [17] and Silverman [18], respectively.
Conflict of Interests
The authors declare that they have no conflict of interests. Please consider this paper for further process.